Scientific Papers - Vi

148 THE COERECTION TO THE LENGTH OF [37
From the differential equation satisfied by J0 and </> we get
. cvt/ o / ix/vf/ w/t/ o 7
d*T J a J a w w
and A2 f /„ (&*) 9 (Ar) r dr = ~ r. /0. -^ + I -^-^rdr;
J a L a? _U •>» ""' a/
so that
r& j" ddb dJo ,~15
(kz — A2) J0 (kr) (/> (A?1) ?" «?• = r,J0 -=r—f -r— 9
J« L rtr *r J«
= — haJ0 (ka) 9' (ha), .........(12)
since here 9 (Act) = 9 (A6) = 0, and also Jtt(kb)=Q. Thus in (11), corr spending to a single term of (8),
R _ ZhaHJp (ka) $' (ha) ,-.~.
The exact .solution demands-the inclusion in (8) of all the admissible vain of A, with addition of (1) which in fact corresponds to a zero value of And each value of A contributes a part to each of the infinite series coefficients B, needed to express the solution on the positive side.
But although an exact solution would involve the whole series of vain of A, approximate methods may be founded upon the use of a limited numb of them. I have used this principle in calculations relating to the potent; from 1870 onwards*. A potential F, given over a closed surface, makes
fAZF\a fdV\* fdV\2\ 7 7 .
1 hr + hr) + hr \\Axdydz,
\\das) \dyj \dz ) J y
reckoned over the whole included volume, a minimum. If an expressi for V, involving a finite or infinite number of coefficients, is proposed whi satisfies the surface condition and is such that it necessarily includes the tr form of V, we may approximate to the value of (14), making it a minimi by variation of the coefficients, even though only a limited number included. Every fresh coefficient that is included renders the approximate closer, and as near an approach as we please to the truth may be arrived by continuing the process. The true value of (14) is equal by Gree theorem to
4-77- JJ dn
(15)
the integration being over the surface, so that at all stages of the appro ination the calculated value of (14) exceeds the true value of (15). In t application to a condenser, whose armatures are at potentials 0 and
* Phil. Trans. Vol. OLXI. p. 77 (1870) ; Scientific Papers, Vol. i. p. 33. Phil. Mag. Vol. XJ p. 328 (1872); Scientific Papers, Vol. i. p. 140. Compare also Phil. Mag. Vol. XLVII. p. (1899), Vol. xxii. p. 225 (1911).< may be applied to the other formulas (9), (10), (11).